L’Hôpital’s rule compares limits of ratios, not their ordinary values.
The numerator and denominator both tend to zero. They are differentiable near zero, and the denominator derivative b is nonzero in this experiment. The derivative ratio has limit a/b, so L’Hôpital’s rule applies. The original quotient remains undefined at x=0.
The rule needs an indeterminate 0/0 or infinity/infinity form and appropriate differentiability conditions.